Presentation at the University of Amsterdam, March 26, 2009.

1. Experiment of Salganik et al. Models Results Conclusions
Social Influence & Popularity
V.A. Traag
March 26, 2009

2. Experiment of Salganik et al. Models Results Conclusions
Outline
1 Experiment of Salganik et al.
2 Models
3 Results
4 Conclusions

3. Experiment of Salganik et al. Models Results Conclusions
Introduction
• What items (e.g. movies, books) become popular?
• Quality leads to popularity? (Harry Potter, Da Vinci code,
Pirandello)
• Idea emerged from web based experiment of Salganik et al.
(Science, 2006)

4. Experiment of Salganik et al. Models Results Conclusions
Experiment of Salganik et al.
• Study inequality and unpredictability experimentally.
• Set up a website with various songs which could be
downloaded.
• Vary some conditions to study the effect of social influence.
• Use multiple realisations to study unpredictability.

5. Experiment of Salganik et al. Models Results Conclusions
Experimental design
More social influence 1
...
More social influence 8
Social influence 1
...
Social influence 8
No social influence 1
...
No social influence 8
User arrival

6. Experiment of Salganik et al. Models Results Conclusions
Screenshots of website

7. Experiment of Salganik et al. Models Results Conclusions
Screenshots of website

8. Experiment of Salganik et al. Models Results Conclusions
Main conclusions
• Inequality rises with social influence.
• Unpredictability rises with social influence.
• Unpredictability also rises with ’quality’.
• Result of a rich-get-richer effect?

9. Experiment of Salganik et al. Models Results Conclusions
BA-model
• Model for links from websites to websites.
• Start out with some small number of websites.
• At each time step add a new website, and add some links.
• Web sites (items) attract links (votes) proportional to the
number of links (votes) (rich-get-richer effect).

10. Experiment of Salganik et al. Models Results Conclusions
BA-model
0
1
2

11. Experiment of Salganik et al. Models Results Conclusions
BA-model
0
1
2
3

12. Experiment of Salganik et al. Models Results Conclusions
BA-model
0
1
2
3
4

13. Experiment of Salganik et al. Models Results Conclusions
BA-model
0
1
2
3
4
5

14. Experiment of Salganik et al. Models Results Conclusions
BA-model
0
1
2
3
4
5
6

15. Experiment of Salganik et al. Models Results Conclusions
BA-model
0
1
2
3
4
5
6
7

16. Experiment of Salganik et al. Models Results Conclusions
BA-model

17. Experiment of Salganik et al. Models Results Conclusions
BA-model
5 10 20 50
0.0050.0200.0500.2000.500
k
Pr(X>k)

18. Experiment of Salganik et al. Models Results Conclusions
BA-model
• We can formalise this process with mathematics.
• Web sites (items) attract links (votes) proportional to the
number of links (votes).
˙ki = m
ki
j kj
• Yields stationary power law degree distribution.
Pr(X = k) = 2m2
k−3

19. Experiment of Salganik et al. Models Results Conclusions
Social influence
• Add a base-line effect of quality.
• Introduce quality φ ≥ 0 with mean quality µ and variance σ.
• Balance quality and popularity through parameter 0 ≤ λ ≤ 1.
• Additional good-get-richer effect.
• New differential equation
˙ki = m (1 − λ)
φi
j φj
+ λ
ki
j kj
.

20. Experiment of Salganik et al. Models Results Conclusions
Theoretical results
0
200
400
600
800
1000
1200
0 100 200 300 400 500
Low Quality
High Quality
High Social Influence
k
t
Time dependent results:
• Votes increase with time
• Older items obtain more
votes
• Better items obtain more
votes
• Changing social influence
changes growth pattern

21. Experiment of Salganik et al. Models Results Conclusions
Theoretical results
Results for items with a given quality
• Mean popularity and variance
E(X|φ) =
mφ
µ
and Var(X|φ) =
E(X|φ)2
1 − 2λ
.
• Expected number of votes rise with quality
• Uncertainty rises with quality and with social influence
• In congruence with experiment from Salganik et al.

22. Experiment of Salganik et al. Models Results Conclusions
Theoretical results
Results for items
• Quality distribution is ρ(φ) with mean µ and variance σ.
• In general, mean popularity and variance is
E(X) = m and Var(X) =
m2(2σ(1 − λ) + µ2)
µ2(1 − 2λ).
• Inequality in popularity increases with inequality in quality
• Inequality rises with social influence
• Again in congruence with experiment from Salganik et al.

23. Experiment of Salganik et al. Models Results Conclusions
Theoretical results
10-30
10-25
10-20
10-15
10-10
10-5
100
100
101
102
103
104
105
106
107
108
109
k
Pr(X=k)
λ = 0
λ = 0.1
λ = 0.5
λ = 0.99

24. Experiment of Salganik et al. Models Results Conclusions
Empirical results
• Quality usually a problem, how to estimate it?
• Workaround: assume a quality distribution (e.g. Dirac,
Exponential).
• Compare empirical popularity distribution (#views, #sales) to
theoretical distribution.
• Estimate social influence parameter λ using MLE.

25. Experiment of Salganik et al. Models Results Conclusions
10
-4
10
-3
10
-2
10
-1
10
0
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
Hollywood
YouTube
Fit (Hollywood)
Fit (YouTube)
k
Pr(X>k)
YouTube1 λ ≈ 0.878
Hollywood1 λ ≈ 0.663 (0.843 for Dirac)
1
Assuming an exponential distribution

26. Experiment of Salganik et al. Models Results Conclusions
Conclusions
Empirical conclusions.
• YouTube shows higher social influence.
• Perhaps a broader distinction (traditional/online)?
• Suggests popular thesis that the Internet individualises is
incorrect.
• With massive choices, following others not a bad heuristic?

27. Experiment of Salganik et al. Models Results Conclusions
Conclusions
Conclusions for model
• Qualitatively congruent with experiment from Salganik.
• Quantitatively not supported by data.
• First rough approximation for modelling the amount of social
influence.
• Might be used for getting rough estimates of social influence.

28. Experiment of Salganik et al. Models Results Conclusions
Questions?